Question

If the volume of a cube is increasing at 24 and the surface area of the cube is increasing at 12, what is the length of each edge of the cube?

Answer 1

Answer:

The length of each edge of the cube is 8 units.            

Step-by-step explanation:

We are given the following in the question:

Let V be the volume of cube, S be the surface area of cube and l be the edge of cube.

$\dfrac{dV}{dt} = 24\\\\\dfrac{dS}{dt} = 12$

We have to find  the length of each edge of the cube.

Volume of cube =

$V = (\text{Edge})^3\\V = l^3\\\\\dfrac{dV}{dt} = 3l^2 \dfrac{dl}{dt}= 24$

Surface area of cube =

$S = 6(\text{Edge})^2\\S = 6l^2\\\\\dfrac{dS}{dt} = 12l \dfrac{dl}{dt}= 12$

Dividing the two equations, we have:

$\dfrac{\frac{dV}{dt}}{\frac{dS}{dt}} = \dfrac{3l^2\frac{dl}{dt}}{12l\frac{dl}{dt}} = \dfrac{24}{12}\\\\\Rightarrow 2 = \dfrac{l}{4}\\\\\Rightarrow l = 8\text{ units}$

The length of each edge of the cube is 8 units.